Problem: $ F = \left[\begin{array}{rr}2 & -1 \\ 0 & 1 \\ 1 & 3\end{array}\right]$ $ A = \left[\begin{array}{rr}3 & 1 \\ 2 & 1\end{array}\right]$ What is $ F A$ ?
Solution: Because $ F$ has dimensions $(3\times2)$ and $ A$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ F A = \left[\begin{array}{rr}{2} & {-1} \\ {0} & {1} \\ \color{gray}{1} & \color{gray}{3}\end{array}\right] \left[\begin{array}{rr}{3} & \color{#DF0030}{1} \\ {2} & \color{#DF0030}{1}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ A$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ A$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ A$ , and so on. Add the products together. $ \left[\begin{array}{rr}{2}\cdot{3}+{-1}\cdot{2} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ A$ and add the products together. $ \left[\begin{array}{rr}{2}\cdot{3}+{-1}\cdot{2} & ? \\ {0}\cdot{3}+{1}\cdot{2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ F$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ A$ and add the products together. $ \left[\begin{array}{rr}{2}\cdot{3}+{-1}\cdot{2} & {2}\cdot\color{#DF0030}{1}+{-1}\cdot\color{#DF0030}{1} \\ {0}\cdot{3}+{1}\cdot{2} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{2}\cdot{3}+{-1}\cdot{2} & {2}\cdot\color{#DF0030}{1}+{-1}\cdot\color{#DF0030}{1} \\ {0}\cdot{3}+{1}\cdot{2} & {0}\cdot\color{#DF0030}{1}+{1}\cdot\color{#DF0030}{1} \\ \color{gray}{1}\cdot{3}+\color{gray}{3}\cdot{2} & \color{gray}{1}\cdot\color{#DF0030}{1}+\color{gray}{3}\cdot\color{#DF0030}{1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}4 & 1 \\ 2 & 1 \\ 9 & 4\end{array}\right] $